tomleslie

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9 years, 243 days

MaplePrimes Activity


These are answers submitted by tomleslie

do a contour plot of a vector field, since the former is magnitude-only and the latter contains both magnitude and direction. You can produce a contour plot of the magnitude of the vector field, which is what I have done in the attached

  restart;
  with(VectorCalculus):
  with(plots):
  epsilon:=.01:
#
# Define the potential and plot it
#
  pot := 1/sqrt(x^2+y^2+epsilon)-1/sqrt((x-.25)^2+y^2+epsilon):
  contourplot(pot, x=-0.5..1, y=-0.5..0.5, grid=[100,100]);
#
# Take the gradient of the potential - which should be the
# electric field
#
  v := VectorField( Gradient( pot, [x,y] ), 'cartesian'[x,y] ):
#
# Produce a field plot
#
  fieldplot(v, x=-0.5..0.5, y=-0.5..0.5, arrows=THICK, color=red, anchor=tail);
#
# OP wants a contour plot of the electric field. However
# a contour plot is magnitude-only. Presumably OP wants the
# contour plot to represent the purely the magnitude of the
# vector field
#
  contourplot(Norm(v)(<x,y>), x=-0.5..1, y=-0.5..0.5, grid=[100,100]);

 

 

 

 


 

Download contplot.mw

which I *think* I have fixed in the attached. This involved removing lots of redundant stuff and a fair amount of editing so the chances that I got everything correct is not that great. I suggest you read/check this very carefully
 

For some reasom this site is not displaying worksheets correctly today, so you will have to download/execute

someODEs.mw

the attached will calculate/plot both speed() and L() for any numeric argument.

Unfortunately the MApleprimes uploader sppears to be broken - again- and will not display the contents of this file on this site so you will have to download/execute

speedandL.mw

My ability to help is limited by the fact I only have a 'bare' copy of Matlab with no addOns/toolboxes, other than the Maple toolbox. However the following observations may be helpful

From the Matlab 'HOME' tab, Add-Ons -> Manage AddOns does not identify the Maple toolbox as a conventional add-on. In fact it does not appear in this list at all. Suggests that one cannot use Matlab'x Add-On manager

On the other hand, from the Matlab 'HOME' tab Set Path shows two Maple-related entries, in my case at the bottom of the list. These entries are

C:\Program Files\MATLAB\R2019b\toolbox\maple
C:\Program Files\MATLAB\R2019b\toolbox\maple\util

So I *assume* that when one installs the Maple toolbox, the Matlab default pathdef.m file (located in C:\Program Files\MATLAB\R2019b\toolbox\local for a "default" installation) is edited to include these "Maple-related" entries. Presumably also, during the Maple toolbox installation, references to the "default" Matlab symbolic toolbox are removed from the pathdef.m file.

So the first thing experiment I would try

  1. Uninstall the Maple toolbox
  2. Verify that the default symbolic toolbox is working
  3. Copy the pathdef.m file to somewhere convenient
  4. Reinstall the Maple toolbox
  5. In a decent editor check the differences between the "current" pathdef.m file and that saved at step (3) above. They *ought* to be different - references to the default symbolic toolbox removed and references to Maple included
  6. At this stage (ie with the Maple toolbox installed and "active"), use the Matlab toolbar HOME->Set Path dialogue to remove the two Maple entries
  7. Path changes take place "immediately" (ie no restart is necessary), this ought to disable the Maple toolbox (without "unistalling" it
  8. Assuming that everything has worked until now find the references to the defalt symbolic toolbox obtained at step 5) above, and again using the HOME->Set Path dialogue, use the Add folder option to reinstate these.
  9. Verify that the default symbolic toolbox is now working
  10. Again assuming that everything is OK, you can now "toggle" beween default and Maple symbolic toolboxes just by using the Add Folder and Remove Folder options in the Set Path dialogue

As has already been noted it is not clear whether you mean 12^(6^5), or (12^6)^5.  The attached does both

  restart:

  doWork:=proc( N::posint)
                local p,
                      num:=N,
                      digsum:=add
                              ( irem
                                (num, 10, 'num'),
                                p=1..length(num)
                              );
                if   irem(digsum, 3)=0
                then printf( "      %a is divisible by 3\n", digsum):
                elif isprime(digsum)
                then printf( "      %a is prime\n", digsum):
                else printf( "      %a is nether prime nor divisible by 3\n")
                fi:
          end proc:
  doWork(12^(6^5));
  doWork((12^6)^5);

      37845 is divisible by 3
      153 is divisible by 3

 

 

 

Download factor.mw

 

You can't use Maple's in-built numeric solver when you have three boundary conditions on the same dependent variable - you have Z(0)=1, Z(1)=exp(1) and Z(3/4)=exp(3/4). In the attached I have used a simple "shooting method" approach to convert the last of these conditions into an additional condition at x=0, by generating an appropriate value for  D[1$2](Z)(0).

Once this has been obtained then you can use Maple's dsolve() with the 'numeric' option to obtain the desired(?) solution

Check the attached

  restart:
  f:= 0:
  Y:= 1+x-58/9*x^2-3*x^2*exp(1)+64/9*x^2*exp(3/4)+40/9*x^3+4*x^3*exp(1)-64/9*x^3*exp(3/4):
  odesys:= [ diff(Z(x), x$4) = 0.001*diff(Y, x$4)+0.999*(f+exp(-x)*Y*Y),
             Z(0) = 1, Z(1)=exp(1),  D(Z)(0) = 1, D[1$2](Z)(0) = icVal]:
  getSol:= proc( alpha )
                 global odesys:
                 if   type(alpha, numeric)
                 then return evalf
                             ( rhs
                               ( dsolve
                                 ( eval
                                   ( odesys,
                                     icVal=alpha
                                   ),
                                   numeric
                                 )(3/4)[2]
                               )-exp(3/4)
                             );
                 else return procname(alpha)
                 fi;
           end proc:
#
# Generate the value for  D[1$2](Z)(0) which will
# "replicate" the BC Z(3/4)=exp(3/4), then solve the
# resulting system
#
  ans:=fsolve( getSol, 0..2):
  sol:=dsolve( eval( odesys, icVal=ans), numeric):
#
# Plot the solution of the ODE with the modified boundary
# condition
#
  p1:= plots:-odeplot( sol, [x, Z(x)], x=0..1);
#
# Plot the difference between the ode solution
# and exp(x) - note that this difference is "zero"
# at x=3/4
#
  p2:= plots:-odeplot( sol, [x, Z(x)-exp(x)], x=0..1);
 

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(10, {(1) = .0, (2) = .1035522254865972, (3) = .24519828381296987, (4) = .3799478193552541, (5) = .503521952794316, (6) = .6168949604764327, (7) = .7212797883990792, (8) = .8185224087372147, (9) = .9105143600838255, (10) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(10, 4, {(1, 1) = 1.0, (1, 2) = 1.0, (1, 3) = .9997284814, (1, 4) = 1.0015789187052664, (2, 1) = 1.109102546816479, (2, 2) = 1.1090838585077165, (2, 3) = 1.1089893945652205, (2, 4) = 1.1105455559517092, (3, 1) = 1.2778701868777473, (3, 2) = 1.2778519409240123, (3, 3) = 1.277938440568671, (3, 4) = 1.2788925273345446, (4, 1) = 1.4622016746038597, (4, 2) = 1.4622014989325756, (4, 3) = 1.4623657325593096, (4, 4) = 1.462544901925881, (5, 1) = 1.6545320253726457, (5, 2) = 1.6545516203064088, (5, 3) = 1.6546924109083185, (5, 4) = 1.6541483740612573, (6, 1) = 1.8531611239783727, (6, 2) = 1.8531919754946065, (6, 3) = 1.8532400609541102, (6, 4) = 1.852184217653946, (7, 1) = 2.057063333053156, (7, 2) = 2.0570928910924984, (7, 3) = 2.057015669943291, (7, 4) = 2.0557095628937962, (8, 1) = 2.2671489949882484, (8, 2) = 2.2671647468270337, (8, 3) = 2.2669574840472175, (8, 4) = 2.265617142608862, (9, 1) = 2.4856028360355884, (9, 2) = 2.485593935859521, (9, 3) = 2.485266381188787, (9, 4) = 2.4839977477406534, (10, 1) = 2.71828182845905, (10, 2) = 2.7182386065361217, (10, 3) = 2.7177993410540364, (10, 4) = 2.716544491925582}, datatype = float[8], order = C_order); YP := Matrix(10, 4, {(1, 1) = 1.0, (1, 2) = .9997284814, (1, 3) = 1.0015789187052664, (1, 4) = .999, (2, 1) = 1.1090838585077165, (2, 2) = 1.1089893945652205, (2, 3) = 1.1105455559517092, (2, 4) = 1.107233907146643, (3, 1) = 1.2778519409240123, (3, 2) = 1.277938440568671, (3, 3) = 1.2788925273345446, (3, 4) = 1.273716128797201, (4, 1) = 1.4622014989325756, (4, 2) = 1.4623657325593096, (4, 3) = 1.462544901925881, (4, 4) = 1.4564700734019, (5, 1) = 1.6545516203064088, (5, 2) = 1.6546924109083185, (5, 3) = 1.6541483740612573, (5, 4) = 1.6487782312719357, (6, 1) = 1.8531919754946065, (6, 2) = 1.8532400609541102, (6, 3) = 1.852184217653946, (6, 4) = 1.8486835183090522, (7, 1) = 2.0570928910924984, (7, 2) = 2.057015669943291, (7, 3) = 2.0557095628937962, (7, 4) = 2.0544306500099254, (8, 1) = 2.2671647468270337, (8, 2) = 2.2669574840472175, (8, 3) = 2.265617142608862, (8, 4) = 2.2660572971002355, (9, 1) = 2.485593935859521, (9, 2) = 2.485266381188787, (9, 3) = 2.4839977477406534, (9, 4) = 2.4848304338285216, (10, 1) = 2.7182386065361217, (10, 2) = 2.7177993410540364, (10, 3) = 2.716544491925582, (10, 4) = 2.715563546630516}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(10, {(1) = .0, (2) = .1035522254865972, (3) = .24519828381296987, (4) = .3799478193552541, (5) = .503521952794316, (6) = .6168949604764327, (7) = .7212797883990792, (8) = .8185224087372147, (9) = .9105143600838255, (10) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(10, 4, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = 0.15174185478711788e-7, (2, 1) = -0.20259265106448437e-9, (2, 2) = 0.8184857433913237e-10, (2, 3) = 0.15700986201749992e-8, (2, 4) = 0.15173052815765837e-7, (3, 1) = -0.7454669674564436e-9, (3, 2) = 0.4616826296409464e-9, (3, 3) = 0.37125208110290578e-8, (3, 4) = 0.15171897592267647e-7, (4, 1) = -0.11247059311640284e-8, (4, 2) = 0.1103849487456392e-8, (4, 3) = 0.57516164716630034e-8, (4, 4) = 0.15173880200643165e-7, (5, 1) = -0.12977351049586068e-8, (5, 2) = 0.19330216455853525e-8, (5, 3) = 0.7622879934494e-8, (5, 4) = 0.15175083684087017e-7, (6, 1) = -0.12958729257339628e-8, (6, 2) = 0.28963748328144248e-8, (6, 3) = 0.9340817724486851e-8, (6, 4) = 0.1517549632148347e-7, (7, 1) = -0.11472396637035051e-8, (7, 2) = 0.39551106747826225e-8, (7, 3) = 0.10923400363554895e-7, (7, 4) = 0.1517554413889154e-7, (8, 1) = -0.8748252483861432e-9, (8, 2) = 0.5089774530152374e-8, (8, 3) = 0.12398199661631365e-7, (8, 4) = 0.15175461093085685e-7, (9, 1) = -0.4920486832962509e-9, (9, 2) = 0.6295037547047582e-8, (9, 3) = 0.13793650783357036e-7, (9, 4) = 0.15175340432771107e-7, (10, 1) = .0, (10, 2) = 0.7590588996645805e-8, (10, 3) = 0.15151223114671366e-7, (10, 4) = 0.15175200116094352e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[10] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(1.517554413889154e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [4, 10, [Z(x), diff(Z(x), x), diff(diff(Z(x), x), x), diff(diff(diff(Z(x), x), x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[10] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[10] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(4, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(10, 4, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(4, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(10, 4, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[Z(x), diff(Z(x), x), diff(diff(Z(x), x), x), diff(diff(diff(Z(x), x), x), x)]'[i] = yout[i], i = 1 .. 4)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[10] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(1.517554413889154e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [4, 10, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[10] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[10] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(10, 4, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(4, {(1) = 0., (2) = 0., (3) = 0., (4) = 0.}); `dsolve/numeric/hermite`(10, 4, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 4)] end proc, (2) = Array(0..0, {}), (3) = [x, Z(x), diff(Z(x), x), diff(diff(Z(x), x), x), diff(diff(diff(Z(x), x), x), x)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[Z(x), diff(Z(x), x), diff(diff(Z(x), x), x), diff(diff(diff(Z(x), x), x), x)]'[i] = res[i+1], i = 1 .. 4)] catch: error  end try end proc

 

 

 

 

 

NULL

Download shoot.mw

Maple and Matlab are two completely different software packages. This has a few consequences

  1. Code which runs in Maple will not run in Matlab
  2. Code which runs in Matlab will not run in Maple

The attached will run in Maple

  restart;
  j:= 500:      tp:= 0.07*j: Ap:= 0.25: Aq:= 0.3:     tq:= 0.03*j:
  tr1:= 0.05*j: Ar:= 2:      As:= 0.4:  tr2:= 0.04*j: ts:= 0.04*j:
  tt:= 0.17*j:  At:= 0.4:

  u1:= 0:
  u2:= (Ap/2)*(sin((2*Pi*t/tp)+(3*Pi/2))+1):
  u3:= 0:
  u4:= -(Aq/tq)*t:
  u5:= (((Ar+ Aq)/tr1)*t)-Aq:
  u6:= -(((Ar+ As)/tr2)*t)+Ar:
  u7:= ((As/ts)*t)- As:
  u8:= 0:
  u9:= (At/2)*(sin((2*Pi*t/tt)+(3*Pi/2))+1):
  u10:= 0:

  with(plots):
  display( [ plot(u1, t=1..0.1*j),
             plot(u2, t=1..tp),
             plot(u3, t=1..0.08*j),
             plot(u4, t=1..tq),
             plot(u5, t=1..tr1),
             plot(u6, t=1..tr2),
             plot(u7, t=1..ts),
             plot(u8, t=1..0.1*j),
             plot(u9, t=1..tt),
             plot(u10,t=1..0.08*j)
           ],
           view=[0..100,-1..3],
           axes=boxed
         );

 

 

Download doPlots.mw

that you actuallly do have the data as a matrix, then

ExportMatrix(fileName, matrixName)

ought to work.

To recall this information

M:=ImportMatrix(fileName)

will retrieve the matrix

is shown in the attached

  restart;
#
# Define a simple function called 'delta'
#
  delta:= (k::integer, r::integer)->`if`( k-r-1=0, 1, 0 ):
#
# A couple of examples
#
  delta(5,4);
  delta(5,3);

1

 

0

(1)

 

Download simpleFunc.mw

well

  1. In the first place I don't see why you are using spline fits, since the process you invoke is
    1. define a function
    2. plot the function
    3. extract data from the plot
    4. fit this data using splines to produce an approximation of the function which you already know at step (1) above
    5. use this approximate "fit" in subsequent calculations. Why????
    6. Hence I have removed all the spline fit stuff - it is pointless in the worksheet you supply
  2. You state that "in graph t=4000 my result is illogical". Actually no -it is entirely logical and has to to with the magnitudes of the functions h(x) and r(t) used in defining the boundary and initial conditions. The first of these is zero unless 5000<x<10000: within this range it never exceeds ~1.2*10-37 so could be considered very, very small with "little impact" on the overall PDE solution. On the other hand the initial condition C(0, t) = r(t) is identically zero for t<3600 and t>21600, but between these values, it gets very big, very quickly. For your first four plots, ie t= 2, 1000, 2000, 3000, C(0,t) is identically zero. For the case you find illogical (ie t=4000), the boundary condition evaluates to C(0,t)=2.88*10^7. I (for one) am not really surprised that when you move from a region where a boundary condition evaluates to 0, to one where the same boundary condition evaluates to 2.88*10^7 then one might expect a significant change in the PDE solution

More details are given in the comments in the attached

  restart;

  with(plots):
  with(plottools):
  with(CurveFitting):

  f:= x -> piecewise( 0 < x and x <= 5000, 0,
                      5000 < x and x <= 7500, 0.008*x - 40,
                      7500 < x and x <= 10000, -0.008*x + 80,
                      10000 < x and x <= 15000, 0
                    );
  Ic:=plot( f(x),
            x = 0 .. 15000,
            color = blue,
            legend = ["Initial Condition"],
            labels = ["x", "concentration"],
            axis = [tickmarks = [5, subticks = 1]]
          );

f := proc (x) options operator, arrow; piecewise(0 < x and x <= 5000, 0, 5000 < x and x <= 7500, 0.8e-2*x-40, 7500 < x and x <= 10000, (-1)*0.8e-2*x+80, 10000 < x and x <= 15000, 0) end proc

 

 

  g := t -> piecewise( 0 < t and t <= 3600, 0,
                       3600 < t and t <= 10800, t/240-15,
                       10800 < t and t <= 21600, -t/360+ 60,
                       21600 < t and t <= 36000, 0
                      );
  Bc:= plot( g(t),
             t = 0 .. 36000,
             color = green,
             legend = ["Boundary Condition"],
             labels = ["time(s)", "concentration"],
             axis = [tickmarks = [10, subticks = 1]]
           );

g := proc (t) options operator, arrow; piecewise(0 < t and t <= 3600, 0, 3600 < t and t <= 10800, (1/240)*t-15, 10800 < t and t <= 21600, -(1/360)*t+60, 21600 < t and t <= 36000, 0) end proc

 

 

  v := 0.5: alpha := 15: mu := (-v)/(2*alpha): beta := v^2/(4*alpha):
  h := x -> f(x)/exp(-mu*x):
  r := t -> g(t)/exp(-beta*t):
#
# As defined above the function f() has a maximum of of 20
# at x=7500. However at this x-value, the value of exp(-mu*x)
# as given by eval(exp(-mu*x), x=7500) will be 1.935576042*10^54
# so that the maximum value of the function h() will be h(7500)
# which comes out to be 1.033284127*10^(-53), which is pretty
# close to zero for all practical purposes
#
# Hence when the OP uses C(x, 0) = h(x) as a boundary condition,
# this is identically zero for x<5000 and x>10000. It is also
# very close to zero in the range 5000<x<10000. The value
# actually "peaks" at around 1.136946279*10^(-37) when x=~5060
# as can be seen in the first of the following plots
#
  plot(h(x), x=5000..10000);
#  
# As defined above the function g() has a maximum of 30, when
# t=10800. However the denominator exp(-beta*t) has a value of
# 2.862518552*10^(-20) at the same t-value. Hence the function
# r() is identically zero for t<3600 and t>21600. But somehere
# around t=10800, it becomes 30/2.862518552*10^(-20) or about
# 1.048028142*10^21. Since the denominator decreases exponentially
# as t increases the value of r() will continue to increase
# beyond the nominal maximum in the function g(), reaching a peak
# of about 2.9*10^38 somewhere around t=21350 as can be seen
# from the second of the following plots
#
  plot(r(t), t=3600..21600);

 

 

#
# Bearing in mind the range of values illustrated above for
# h(x) and r(t), consider the PDE and the BCs/ICs
#
# For any value of t<3600, the boundary condition C(0,t) is
# identically zero. Hence (roughly speaking) the values of
# C(x,t) for t=2,1000,2000, 3000, are governed mainly by the
# the form of the ODE and the BC C(x, 0) = h(x). Since h(x)
# is always "tiny" all of these solutions will be "tiny" as
# seen in the first plot below
#
# However for t>3600 (as in the plot case t=4000), the value
# of C(0,t)=r(t) becomes "huge" (r(4000)=2.88*0^7) with a
# consequent effect on the PDE solution as seen in the second
# plot below
#
  PDED := diff(C(x, t), t) = 15*diff(C(x, t), x, x):
  IBCD := C(x, 0) = h(x), C(0, t) = r(t), D[1](C)(15000, t) - mu*C(15000, t) = 0:
  pdsD := pdsolve(PDED, [IBCD], time = t, range = 0 .. 15000):
  display( [ pdsD:-plot(C(x,t), t = 2,    color = red),
             pdsD:-plot(C(x,t), t = 1000, color = blue),
             pdsD:-plot(C(x,t), t = 2000, color = green),
             pdsD:-plot(C(x,t), t = 3000, color = orange)
           ]
         );
  pdsD:-plot(C(x,t), t = 4000, color = black);

 

 

 

Download pdeProb.mw

in the past, is usually associated with a bootleg/cracked copy of Maple

If your copy is legit then Technical Support at Maplesoft.com will fix it.

  1. Most non-lineear ODEs don't have  analyic solutions, despite the impression given by your textbook
  2. The solution of any non-linear ODE (actually an linear ODE as well) is highly dependent on the values of any parameters and initial boundary conditions. Your exaqmaple provides no initial conditions (so I made them up) and no parameter values (other than ranges), so I made these up as well. The attached shows a couple of very different "solutions", depending on exactly which values you pick for parameters/ICs

ode := 0 = diff(y(x), x) + ((r + 2*x)*(p - y(x)^(-s)))/(-(b*y(x) - x^2)*s*y(x)^(-s - 1));

#parameters(0 < r, 0 < p, b < 1 and 0 < b, 0 < s)
sol:=dsolve([ode,y(0)=0.1], numeric, parameters=[ r, p, b, s],maxfun=0);

0 = diff(y(x), x)-(r+2*x)*(p-y(x)^(-s))/((b*y(x)-x^2)*s*y(x)^(-s-1))

 

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [r = r, p = p, b = b, s = s]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 1, (2) = 1, (3) = 0, (4) = 0, (5) = 4, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 0, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..5, {(1) = .1, (2) = Float(undefined), (3) = Float(undefined), (4) = Float(undefined), (5) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8]), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; if Y[1] < 0 then YP[1] := undefined; return 0 end if; if Y[1] < 0 then YP[1] := undefined; return 0 end if; YP[1] := (Y[2]+2*X)*(Y[3]-Y[1]^(-Y[5]))/((-X^2+Y[1]*Y[4])*Y[5]*Y[1]^(-Y[5]-1)); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; if Y[1] < 0 then YP[1] := undefined; return 0 end if; if Y[1] < 0 then YP[1] := undefined; return 0 end if; YP[1] := (Y[2]+2*X)*(Y[3]-Y[1]^(-Y[5]))/((-X^2+Y[1]*Y[4])*Y[5]*Y[1]^(-Y[5]-1)); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..5, {(1) = 0., (2) = .1, (3) = undefined, (4) = undefined, (5) = undefined}); _vmap := array( 1 .. 1, [( 1 ) = (1)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, y(x)], (4) = [r = r, p = p, b = b, s = s]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(1)

sol(parameters=[0.1, 2.0, 10.0, 0.3]):
plots:-odeplot( sol, [x,y(x)], x=0..10 );

 

sol(parameters=[0.1, 2.0, 10.0, 0.4]):
plots:-odeplot( sol, [x,y(x)], x=0..10 );

Warning, cannot evaluate the solution further right of .64031848, probably a singularity

 

 

 

Download nonlinODE.mw

 

Exactly what is supposed to be "wrong" with this worksheet

The atached shows your worksheet running in Maple 18. The only problems arise becuase of the inclusion of the commands

set 1;

at various points. These errors do not surprise me, since this command obeys no Maple syntax and is in fact completely meaningless. Other than your use of meaningless commands, what else is supposed to be wrong with the attached??

restart:
interface(version);

`Standard Worksheet Interface, Maple 18.02, Windows 7, October 20 2014 Build ID 991181`

(1)

restart;
P := -lambda*exp(-Phi(xi))-mu*exp(Phi(xi));
u[0] := A[0]+A[1]*exp(-Phi(xi))+A[2]*exp(-Phi(xi))*exp(-Phi(xi));
u[1] := diff(u[0], xi);
d[1] := -A[1]*P*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*P;
d[2] := -A[1]*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)))*exp(-Phi(xi))+A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P*exp(-Phi(xi))+4*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P-2*A[2]*(exp(-Phi(xi)))^2*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)));
collect(expand((2*k*k)*w*beta*d[2]-(2*alpha*k*k)*d[1]-2*w*u[0]+k*u[0]*u[0]), exp(Phi(xi)));

-lambda*exp(-Phi(xi))-mu*exp(Phi(xi))

 

A[0]+A[1]*exp(-Phi(xi))+A[2]*(exp(-Phi(xi)))^2

 

-A[1]*(diff(Phi(xi), xi))*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*(diff(Phi(xi), xi))

 

-A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))

 

-A[1]*(lambda*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-mu*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(Phi(xi)))*exp(-Phi(xi))+A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))^2*exp(-Phi(xi))+4*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))^2-2*A[2]*(exp(-Phi(xi)))^2*(lambda*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-mu*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(Phi(xi)))

 

4*k^2*w*beta*A[2]*mu^2-2*alpha*k^2*A[1]*mu+k*A[0]^2-2*w*A[0]+(4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1])/exp(Phi(xi))+(16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2])/(exp(Phi(xi)))^2+(4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2])/(exp(Phi(xi)))^3+(12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2)/(exp(Phi(xi)))^4

(2)

restart;
solve({12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2, 4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2], 4*beta*k^2*mu^2*w*A[2]-2*alpha*k^2*mu*A[1]+k*A[0]^2-2*w*A[0], 4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1], 16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2]}, {k, w, A[0], A[1], A[2]});

set 1;

{k = 0, w = 0, A[0] = A[0], A[1] = A[1], A[2] = A[2]}, {k = k, w = w, A[0] = 0, A[1] = 0, A[2] = 0}, {k = k, w = w, A[0] = 2*w/k, A[1] = 0, A[2] = 0}, {k = RootOf(24*beta*lambda*mu*_Z^2-1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (1/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2-1)), A[1] = (1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2-1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2-1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}, {k = RootOf(24*beta*lambda*mu*_Z^2+1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (3/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2+1)), A[1] = -(1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2+1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2+1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}

 

Error, unexpected number

 

restart;
solve({12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2, 4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2], 4*beta*k^2*mu^2*w*A[2]-2*alpha*k^2*mu*A[1]+k*A[0]^2-2*w*A[0], 4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1], 16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2]}, {k, w, A[0], A[1], A[2]});

set 1;

{k = 0, w = 0, A[0] = A[0], A[1] = A[1], A[2] = A[2]}, {k = k, w = w, A[0] = 0, A[1] = 0, A[2] = 0}, {k = k, w = w, A[0] = 2*w/k, A[1] = 0, A[2] = 0}, {k = RootOf(24*beta*lambda*mu*_Z^2-1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (1/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2-1)), A[1] = (1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2-1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2-1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}, {k = RootOf(24*beta*lambda*mu*_Z^2+1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (3/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2+1)), A[1] = -(1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2+1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2+1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}

 

Error, unexpected number

 

restart;
solve({24*Z^2*beta*lambda*mu-1}, {Z});
solve({100*Z^2*lambda*mu+1}, {Z});

{Z = (1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)}, {Z = -(1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)}

 

{Z = -(1/10)/(-lambda*mu)^(1/2)}, {Z = (1/10)/(-lambda*mu)^(1/2)}

(3)

restart;
k := (1/12)*sqrt(6)/sqrt(beta*lambda*mu);
w := -alpha/((10*sqrt(-lambda*mu))*beta);
A[0] := 1/2*(-alpha/((10*sqrt(-lambda*mu))*((1/12)*beta*sqrt(6)/sqrt(beta*lambda*mu))));
A[1] := (1/10)*alpha/((1/12)*beta*mu*sqrt(6)/sqrt(beta*lambda*mu));
A[2] := (12*(1/12))*sqrt(6)*lambda^2*alpha/(sqrt(beta*lambda*mu)*(10*sqrt(-lambda*mu)));
lambda := 3;
mu := 2;
H := -ln(sqrt(lambda/mu)*tan(sqrt(lambda*mu)*(xi+C)));
u[0] := A[0]+A[1]*exp(-H)+A[2]*exp(-H)*exp(-H);
f := diff(u[0], xi);
S := diff(f, xi);
simplify(%);

(1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)

 

-(1/10)*alpha/((-lambda*mu)^(1/2)*beta)

 

-(1/10)*alpha*6^(1/2)*(beta*lambda*mu)^(1/2)/((-lambda*mu)^(1/2)*beta)

 

(1/5)*alpha*6^(1/2)*(beta*lambda*mu)^(1/2)/(beta*mu)

 

(1/10)*6^(1/2)*lambda^2*alpha/((beta*lambda*mu)^(1/2)*(-lambda*mu)^(1/2))

 

3

 

2

 

-ln((1/2)*6^(1/2)*tan(6^(1/2)*(xi+C)))

 

(1/10)*alpha*(-6)^(1/2)/beta^(1/2)+(3/10)*alpha*6^(1/2)*tan(6^(1/2)*(xi+C))/beta^(1/2)-(9/40)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))^2/beta^(1/2)

 

(9/5)*alpha*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)-(9/20)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))*6^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)

 

(18/5)*alpha*tan(6^(1/2)*(xi+C))*6^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)-(27/10)*alpha*(-6)^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)^2/beta^(1/2)-(27/5)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))^2*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)

 

(9/10)*alpha*6^(1/2)*((6*I)*cos(6^(1/2)*(xi+C))^2+4*sin(6^(1/2)*(xi+C))*cos(6^(1/2)*(xi+C))-9*I)/(cos(6^(1/2)*(xi+C))^4*beta^(1/2))

(4)

 

Download satisfy18.mw

 

are shown in the attached

  d1:= [[3, 11], [4, 6], [5, 8]]:
#
# The 'quick' way - direct calculatiom
#
  add(i[1]-2, i in d1);

6

(1)

#
# If you really want a procedure/function -
# probably only worth it if the process is
# being applied to several input lists.
#
# Define the function
#
  f:=lis-> local i:
           add( i[1]-2, i in lis):
#
# Apply the function to the argument
#
  f(d1);

6

(2)

 

Download addCoord.mw

In the attached, I have made a few adjustments

  1. I don't have access to Maple's GlobalOptimization  add-on - so this code section is commented out. Although the best answer achieved (ie 6.979) should be borne in mind for future reference.
  2. I rewrote the code around the DirectSearch() section - just so that it would run, there were various syntax issues! The result isn't pretty but it works
    1. Using the GlobalOptima() command from the DirectSearch package produce a "maximum" of 3.95463824386087 - so obviously not as "maximal" as the value obtained in (1) above
    2. Rather than using DirectSearch:-GlobalOptima(), I tried defining 20 different initial points (you can change this number via N__inits), just to see how well a basic DirectSearch:-Search() command would do, running from different start points. This procuced a "maximum" of 4.17769689355030. So better than that obtained form DirectSearch:-GlobalOptima, but not as good as the GlobalOptimization add-on
  3. I think this variable behaviour really just illustrates the whole issue of optimization for non-convex functions. There is no known algorithm which is guaranteed to find the optimal solution. Nearly all of the approaches I am familiar with perform the following steps (for maximization)
    1. Guess an initial point
    2. Do a few evaluations of the objective function around this point to figure out which way is "uphill"
    3. Step "uphill"
    4. Repeat steps (2)-(3) untial nothing is "uphill" any more - this is a "local" maximum
    5. Guess other initial points and repeat (2)-(3) above, retaining that which provides the highest "local" maximum
    6. Make the (somewhat rash) statement, that the "highest" of all the "local" maxima obtained is in fact the "global" maxima
    7. The success/failure of this strategy is almost completely governed by the number and selection of the initial points. Covering a ssufficient(?) number of initial points depends on the number of variables in the objecive function and what sort of "spacing/range" of intial points makes sense. The latter criterion is an "art" not a "science"
  4. MUch more interesting (to me at least) is the final execution group in the original worksheet, which by a "coordinate transformation" changes the problem to a linear program which can be solve quickly, and provides the same solution as Maple's GlobalOptimization add-on. This is impressive, and I'm still trying to get my head around why it works. If I were you, this is where I would focus my attention

Anyhow, for what it is worth, and bearing in mind all of the caveats above, see the attached

Portfolio Optimization with the Omega Ratio

 

Introduction

 

 

Traditional investment performance benchmarks, like the Sharpe Ratio, approximate the returns distribution with mean and standard deviation. This, however, assumes the distribution is normal.  Many modern investments vehicles, like hedge funds, display fat tails, and skew and kurtosis in the returns distribution. Hence, they cannot be adequately benchmarked with traditional approaches.

 

One solution, proposed by Shadwick and Keating in 2002 is the Omega Ratio.  This divides the returns distribution into two halves – the area below a target return, and the above a target return. The Omega Ratio is simply the former divided by the latter. A higher value is better.

 

For a set of discrete returns, the Omega Ratio is given by

 

Omega(L) = E[max(R-L, 0)]/E[max(L-R, 0)]

 

where L is a target return and R is a vector of returns.

 

This application finds the asset weights that maximize the Omega Ratio of a portfolio of ten investments, given their simulated monthly returns and a target return.

 

This is a non-convex problem, and requires global optimizers for a rigorous solution. However, a transformation of the variables (only valid for Omega Ratios of over 1) converts the optimization into a linear program.

 

This application implements both approaches, the former using Maple's Global Optimization Toolbox, and the latter using Maple's linear programming features. For the data set provided in this application, both approaches give comparible results.

 

Returns Data and Minimum Acceptable Return

 

restart

Monthly hedge fund returns

Number of funds

N := LinearAlgebra[ColumnDimension](data)

10

(2.1)

Number of returns for each fund

S := LinearAlgebra[RowDimension](data)

36

(2.2)

Target Return

L := .1

Omega Ratio

 

OmegaRatio := proc (L, returns, weights) local weightedReturns, above, below, N, S, a, i, j; if convert(`~`[type](weights, numeric), set)[] then N := LinearAlgebra[ColumnDimension](returns); S := LinearAlgebra[RowDimension](returns); weightedReturns := [seq(add(returns[i, j]*weights[j], j = 1 .. N), i = 1 .. S)]; below := select(proc (x) options operator, arrow; evalb(x <= L) end proc, weightedReturns); above := select(proc (x) options operator, arrow; evalb(L < x) end proc, weightedReturns); return add(a-L, `in`(a, above))/add(L-a, `in`(a, below)) else return 'procname(args)' end if end proc

 

"Strawman" portfolio of equal weights at the target return

OmegaRatio(L, data, [.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])

HFloat(4.003192510007608)

(2.3)

Global Optimization

 

NULL

 

Optimized Omega Ratio

NULL

Optimized investment weights

NULL

Direct Search

 

with(DirectSearch); wrapper := proc () global L, data; OmegaRatio(L, data, [args]) end proc; ansGlobal := GlobalOptima(wrapper(seq(w[i], i = 1 .. 10)), [add(w[i], i = 1 .. N) = 1, seq(w[i] = 0 .. .1, i = 1 .. N)], maximize); r := rand(0. .. 1.0); N__inits := 20; for j to N__inits do rvals := seq(r(), j = 1 .. N); ans[j] := Search(wrapper(seq(w[i], i = 1 .. N)), [add(w[i], i = 1 .. N) = 1, seq(w[i] = 0 .. .1, i = 1 .. N)], initialpoint = [seq(w[i] = `~`[`/`]([rvals], add(rvals))[i], i = 1 .. N)], usewarning = false, maximize) end do; seq(ans[j][1], j = 1 .. N__inits); ans[max[index]([seq(ans[j][1], j = 1 .. N__inits)])]

[HFloat(3.954638243860872), [w[1] = HFloat(0.1), w[2] = HFloat(0.1), w[3] = HFloat(0.09714257075732306), w[4] = HFloat(0.09710635877067376), w[5] = HFloat(0.09968784671681309), w[6] = HFloat(0.09353148613677527), w[7] = HFloat(0.09999450830780217), w[8] = HFloat(0.09999999999998163), w[9] = HFloat(0.0806020956977236), w[10] = HFloat(0.09959361372477742)], 1780]

 

HFloat(3.865771689021921), HFloat(4.023426840990234), HFloat(3.7192361069502007), HFloat(3.995249533082313), HFloat(3.9377227987097942), HFloat(3.929660158669612), HFloat(3.976521807439104), HFloat(3.8353678058307317), HFloat(4.177696893550296), HFloat(3.905839821607468), HFloat(3.8932377942132868), HFloat(3.955000171243558), HFloat(3.8678107038369087), HFloat(3.7031708987131884), HFloat(3.867738407138106), HFloat(3.879604472834744), HFloat(3.989758201454413), HFloat(3.7912897819280573), HFloat(4.016230388564598), HFloat(3.9414606304823896)

 

[HFloat(4.177696893550296), [w[1] = HFloat(0.09565526251705606), w[2] = HFloat(0.09996201616302475), w[3] = HFloat(0.045971685565773905), w[4] = HFloat(0.09896111064147), w[5] = HFloat(0.09797375100631699), w[6] = HFloat(0.09999918334598228), w[7] = HFloat(0.1), w[8] = HFloat(0.07669946978292375), w[9] = HFloat(0.1), w[10] = HFloat(0.1)], 2276]

(2.3.1)

Linear Program

 

 

The transformation of the optimization problem into a linear program is described here

eq1 := seq(add(data[i, j]*w[j], j = 1 .. N)-u[i]+d[i]-L*t = 0, i = 1 .. S)

eq2 := add(w[j], j = 1 .. N) = t

eq3 := add(d[i]/S, i = 1 .. S) = 1

obj := add(u[i]/S, i = 1 .. S)

cons := seq([u[i] >= 0, d[i] >= 0][], i = 1 .. S), seq([w[j] >= 0][], j = 1 .. N)

resultsLP := Optimization[LPSolve](obj, {cons, eq1, eq2, eq3}, maximize, assume = nonnegative)


Optimized Omega Ratio

resultsLP[1]

6.97971754550025

(2.4.1)

Optimized investment weights

assign(select(has, resultsLP[2], t))

weightsLP := map(proc (i) options operator, arrow; lhs(i) = rhs(i)/t end proc, select(has, resultsLP[2], w))

[w[1] = HFloat(0.0), w[2] = HFloat(0.0), w[3] = HFloat(0.0), w[4] = HFloat(0.40979022216167177), w[5] = HFloat(0.0), w[6] = HFloat(0.21961738132814781), w[7] = HFloat(0.33321118343926637), w[8] = HFloat(0.0), w[9] = HFloat(0.0), w[10] = HFloat(0.037381213348692446)]

(2.4.2)

 

 

Download optProb.mw

 

 

 

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